Theorem relcmpcmet 25360

Description: If 𝐷 is a metric space such that all the balls of some fixed size are relatively compact, then 𝐷 is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)

Hypotheses

Ref	Expression
relcmpcmet.1	⊢ 𝐽 = (MetOpen‘𝐷)
relcmpcmet.2	⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
relcmpcmet.3	⊢ (𝜑 → 𝑅 ∈ ℝ+)
relcmpcmet.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)

Assertion

Ref	Expression
relcmpcmet	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))

Distinct variable groups:   𝑥,𝐷   𝑥,𝐽   𝜑,𝑥   𝑥,𝑅   𝑥,𝑋

Proof of Theorem relcmpcmet

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcmpcmet.2	. 2 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
2		metxmet 24374	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
4	3	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝐷 ∈ (∞Met‘𝑋))
5		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (CauFil‘𝐷))
6		relcmpcmet.3	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ+)
7	6	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑅 ∈ ℝ+)
8		cfil3i 25311	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘𝐷) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
9	4, 5, 7, 8	syl3anc 1389	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
10	3	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐷 ∈ (∞Met‘𝑋))
11		relcmpcmet.1	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘𝐷)
12	11	mopntopon 24479	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
13	10, 12	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
14		cfilfil 25309	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (Fil‘𝑋))
15	3, 14	sylan 589	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (Fil‘𝑋))
16	15	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
17		simprr 782	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
18		topontop 22953	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
19	13, 18	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐽 ∈ Top)
20		simprl 780	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑥 ∈ 𝑋)
21	6	rpxrd 13035	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℝ*)
22	21	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑅 ∈ ℝ*)
23		blssm 24458	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝑋)
24	10, 20, 22, 23	syl3anc 1389	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝑋)
25		toponuni 22954	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
26	13, 25	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑋 = ∪ 𝐽)
27	24, 26	sseqtrd 3972	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ ∪ 𝐽)
28		eqid 2761	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
29	28	clsss3 23099	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ ∪ 𝐽)
30	19, 27, 29	syl2anc 593	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ ∪ 𝐽)
31	30, 26	sseqtrrd 3973	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋)
32	28	sscls 23096	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ∪ 𝐽) → (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
33	19, 27, 32	syl2anc 593	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
34		filss 23893	. . . . . . . 8 ⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ ((𝑥(ball‘𝐷)𝑅) ∈ 𝑓 ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓)
35	16, 17, 31, 33, 34	syl13anc 1390	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓)
36		fclsrest 24064	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) = ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
37	13, 16, 35, 36	syl3anc 1389	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) = ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
38		inss1 4188	. . . . . . 7 ⊢ ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ⊆ (𝐽 fClus 𝑓)
39		eqid 2761	. . . . . . . . 9 ⊢ dom dom 𝐷 = dom dom 𝐷
40	11, 39	cfilfcls 25316	. . . . . . . 8 ⊢ (𝑓 ∈ (CauFil‘𝐷) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
41	40	ad2antlr 737	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
42	38, 41	sseqtrid 3978	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ⊆ (𝐽 fLim 𝑓))
43	37, 42	eqsstrd 3970	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ⊆ (𝐽 fLim 𝑓))
44		relcmpcmet.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
45	44	ad2ant2r 757	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
46		filfbas 23888	. . . . . . . . . 10 ⊢ (𝑓 ∈ (Fil‘𝑋) → 𝑓 ∈ (fBas‘𝑋))
47	16, 46	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑓 ∈ (fBas‘𝑋))
48		fbncp 23879	. . . . . . . . 9 ⊢ ((𝑓 ∈ (fBas‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓) → ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓)
49	47, 35, 48	syl2anc 593	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓)
50		trfil3 23928	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋) → ((𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ↔ ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓))
51	16, 31, 50	syl2anc 593	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ↔ ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓))
52	49, 51	mpbird 259	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
53		resttopon 23201	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
54	13, 31, 53	syl2anc 593	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
55		toponuni 22954	. . . . . . . . 9 ⊢ ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) = ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
56	54, 55	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) = ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
57	56	fveq2d 6867	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) = (Fil‘∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))))
58	52, 57	eleqtrd 2863	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))))
59		eqid 2761	. . . . . . 7 ⊢ ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) = ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
60	59	fclscmpi 24069	. . . . . 6 ⊢ (((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp ∧ (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅)
61	45, 58, 60	syl2anc 593	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅)
62		ssn0 4357	. . . . 5 ⊢ ((((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ⊆ (𝐽 fLim 𝑓) ∧ ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅) → (𝐽 fLim 𝑓) ≠ ∅)
63	43, 61, 62	syl2anc 593	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 fLim 𝑓) ≠ ∅)
64	9, 63	rexlimddv 3168	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝑓) ≠ ∅)
65	64	ralrimiva 3153	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)
66	11	iscmet 25326	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
67	1, 65, 66	sylanbrc 592	1 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  ∃wrex 3085   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  ∪ cuni 4864  dom cdm 5645  ‘cfv 6517  (class class class)co 7392  ℝ^*cxr 11212  ℝ⁺crp 12990   ↾_t crest 17432  ∞Metcxmet 21389  Metcmet 21390  ballcbl 21391  fBascfbas 21392  MetOpencmopn 21394  Topctop 22933  TopOnctopon 22950  clsccl 23058  Compccmp 23426  Filcfil 23885   fLim cflim 23974   fClus cfcls 23976  CauFilccfil 25294  CMetccmet 25296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147  ax-pre-sup 11148

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-iin 4951  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-er 8673  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-fi 9354  df-sup 9385  df-inf 9386  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-div 11842  df-nn 12208  df-2 12277  df-n0 12479  df-z 12566  df-uz 12837  df-q 12947  df-rp 12991  df-xneg 13111  df-xadd 13112  df-xmul 13113  df-ico 13352  df-rest 17434  df-topgen 17455  df-psmet 21396  df-xmet 21397  df-met 21398  df-bl 21399  df-mopn 21400  df-fbas 21401  df-fg 21402  df-top 22934  df-topon 22951  df-bases 22986  df-cld 23059  df-ntr 23060  df-cls 23061  df-nei 23138  df-cmp 23427  df-fil 23886  df-flim 23979  df-fcls 23981  df-cfil 25297  df-cmet 25299

This theorem is referenced by:  cmpcmet  25361  cncmet  25364